D-mesons in asymmetric nuclear matter Amruta Mishra∗ and Arindam Mazumdar†
arXiv:0810.3067v3 [nucl-th] 23 Feb 2009
Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi – 110 016, India
Abstract ¯ We calculate the in-medium D and D-meson masses in isospin aysmmetric nuclear matter in an ¯ - mass modifications arising from their interactions with the effective chiral model. The D and D nucleons and the scalar mesons in the effective hadronic model are seen to be appreciable at high densities and have a strong isospin dependence. These mass modifications can open the channels ¯ pairs in the dense hadronic matter. of the decay of the charmonium states (Ψ′ , χc , J/Ψ) to D D The isospin asymmetry in the doublet D = (D 0 , D+ ) is seen to be particularly appreciable at high densities and should show in observables like their production and flow in asymmetric heavy ion collisions in the compressed baryonic matter experiments in the future facility of FAIR, GSI. ¯ in-medium masses in The results of the present work are compared to calculations of the D(D) the literature using the QCD sum rule approach, quark meson coupling model, coupled channel approach as well as from the studies of quarkonium dissociation using heavy quark potentials from lattice QCD at finite temperatures. PACS numbers: 24.10.Cn; 13.75.Jz; 25.75.-q
∗
Electronic address:
[email protected],
[email protected]
†
Electronic address:
[email protected],
[email protected]
1
I.
INTRODUCTION
The in-medium properties of hadrons is a topic of intense research in high energy physics since the last few decades. The ongoing and future relativistic heavy ion collision experiments, at the high energy accelerators SPS, CERN, Switzerland; SIS, GSI, Germany; RHIC, BNL, USA; LHC, CERN, Switzerland, and the compressed baryonic matter (CBM) experiments at the future facilities at GSI, Germany, are intended to probe matter at high temperatures and densities. The hadrons modified in the hot and dense hadronic medium resulting from heavy ion collision experiments, affect the experimental observables. E.g.,the dilepton spectra observed from heavy ion collision experiments at the SPS [1, 2] are attributed to the medium modifications of vector mesons [3, 4, 5, 6, 7], and can not be explained by vacuum hadronic properties. The production of kaons and antikaons as well as their collective flow are attributed to the modifications of their spectral functions in the medium [4, 8, 9, 10, 11, 12, 13, 14]. Due to the presence of the light quark/antiquark, ¯ mesons can be appreciable in the dense hadronic the medium modifications of the D (D) medium [15, 16, 17, 18, 19, 20, 21]. The observation of open charm enhancement in nuclear collisions [22, 23] as well as J/Ψ suppression as observed at the SPS [24, 25, 26], can be due ¯ in the medium. In higher energy heavy ion collision to medium modifications of the D (D) experiments at RHIC as well as at LHC, the J/Ψ suppression can arise from formation of a quark-gluon plasma (QGP) [27, 28]. However, the effect of the hadron absorption of J/Ψ ¯ pair in the medium, the is not negligible [29, 30, 31]. With the drop in the mass of D D ¯ final excited states of J/Ψ, a major source of yield of J/Ψ [32], can decay to the D(D) states [33], thereby leading to J/Ψ suppression [34] in the hadronic medium. The medium modification of the masses of J/Ψ and their excited states in the hadronic medium, due to the D-meson mass modifications have also been considered in the literature [35] and the excited states of JΨ are seen to have appreciable mass dop in the medium. The effects due to the mass modifications of the D-mesons as well as the charmonium bound states on experimental observables could be explored at the future accelerator facility at GSI [36]. It is thus important to study the modifications of the charmed mesons in the medium and hence one has to understand the charmed meson interactions in the hadronic phase. In the QCD sum rule approach, due to the presence of the light quark/antiquark, the 2
mass modification of the D-mesons arises from the light quark condensate [15, 16, 19]. In the quark meson coupling (QMC) model, the contribution from the mc h¯ qqiN term is represented by a quark-σ meson coupling. The QMC model predicts the mass shift of the D-meson to be of the order of 60 MeV at nuclear matter density [21], which is very similar to the value obtained in the QCD sum rule calculations of Ref. [15, 19]. Furthermore, lattice calculations for heavy quark potentials at finite temperature suggest a similar drop [20, 37]. In this work we study the medium modification of the masses of open charm mesons (D± ) due to their interactions with the isoscalar-scalar mesons (σ and ζ mesons), isovectorscalar δ mesons and the nucleons in asymmetric nuclear matter. The medium modification of the properties of kaons and antikaons in (isospin asymmetric) dense nuclear matter has been studied in a chiral SU(3)-flavor model in Ref. [38, 39] and have been extended to include the effects from hyperons in the asymmetric strange hadronic matter in ref. [40]. We generalise the model to SU(4)-flavor to derive the interactions of the D mesons with the nucleons and scalar mesons and investigate their mass modifications in the asymmetric matter. The masses of the D ± mesons in symmetric nuclear matter have been studied earlier in such an effective chiral model [41]. In a coupled channel approach for the study of D mesons, using a separable potential, it was shown that the resonance Λc (2593) is generated dynamically in the I=0 channel [42] analogous to Λ(1405) in the coupled channel approach ¯ for the KN interaction [43]. The approach has been generalized to study the spectral density of the D-mesons at finite temperatures and densitites [44], taking into account the modifications of the nucleons in the medium. The results of this investigation seem to indicate a dominant increase in the width of the D-meson whereas there is only a very small change in the D-meson mass in the medium [44]. However, these calculations [42, 44], assume the interaction to be SU(3) symmetric in u,d,c quarks and ignore channels with charmed hadrons with strangeness. A coupled channel approach for the study of D-mesons has been developed based on SU(4) symmetry [45] to construct the effective interaction between pseudoscalar mesons in a 16-plet with baryons in 20-plet representation through exchange of vector mesons and with KSFR condition [46]. This model [45] has been modified in aspects like regularization method and has been used to study DN interactions in Ref. [47]. This reproduces the resonance Λc (2593) in the I=0 channel and in addition generates
3
another resonance in the I=1 channel at around 2770 MeV. These calculations have been generalized to finite temperatures [48] accounting for the in-medium modifications of the ¯ properties [49] in the hot nucleons in a Walecka type σ − ω model, to study the D and D and dense hadronic matter. ¯ enWithin the effective chiral model considered in the present investigation, the D(D) ergies are modified due to a vectorial Weinberg-Tomozawa, due to scalar exchange terms (σ, ζ, δ) as well as range terms [39, 40]. The isospin asymmetric effects among D 0 and D + ¯ ≡ (D¯0 , D − ) in the doublet, D≡ (D 0 , D + ) as well as between D¯0 and D − in the doublet, D arise due to the scalar-isovector δ meson, due to asymmetric contributons in the WeinbergTomozawa term, as well as in the range term [38]. We organize the paper as follows: We briefly recapitulate the SU(3)-flavor chiral model adopted for the description of the asymmetric hadronic matter [39, 40] in Section II. The properties then are studied within this approach. These give rise to medium modifications for the D-masses through their interactions with the nucleons and scalar mesons as presented in Section III. Section IV discusses the results of the present investigation, while we summarise our findings and discuss possible outlook in Section V.
II.
THE HADRONIC CHIRAL SU (3) × SU (3) MODEL
The effective hadronic chiral Lagrangian used in the present work is given as L = Lkin +
X
W =X,Y,V,A,u
LBW + Lvec + L0 + LSB
(1)
are discussed. Eq. (1) corresponds to a relativistic quantum field theoretical model of baryons and mesons adopting a nonlinear realization of chiral symmetry [50, 51, 52] and broken scale invariance (for details see [53, 54, 55]) as a description of the hadronic matter. The model was used successfully to describe nuclear matter, finite nuclei, hypernuclei and neutron stars. The Lagrangian contains the baryon octet, the spin-0 and spin-1 meson multiplets as the elementary degrees of freedom. In Eq. (1), Lkin is the kinetic energy term, LBW contains the baryon-meson interactions in which the baryon-spin-0 meson interaction terms generate the baryon masses. Lvec describes the dynamical mass generation of the vector mesons via couplings to the scalar fields and contains additionally quartic self4
interactions of the vector fields. L0 contains the meson-meson interaction terms inducing the spontaneous breaking of chiral symmetry as well as a scale invariance breaking logarithmic potential. LSB describes the explicit chiral symmetry breaking. The baryon-scalar meson interactions generate the baryon masses and the parameters corresponding to these interactions are adjusted so as to obtain the baryon masses as their experimentally measured vacuum values. For the baryon-vector meson interaction terms, there exist the F -type (antisymmetric) and D-type (symmetric) couplings. Here we will use the antisymmetric coupling [39, 40, 53] because, following the universality principle [56] and the vector meson dominance model, one can conclude that the symmetric coupling should be small. Additionally we choose the parameters [38, 53] so as to decouple the strange vector field φµ ∼ s¯γµ s from the nucleon, corresponding to an ideal mixing between ω and φ. A small deviation of the mixing angle from the ideal mixing [57, 58, 59] has not been taken into account in the present investigation. The Lagrangian densities corresponding to the interaction for the vector meson, Lvec , the meson-meson interaction L0 and that corresponding to the explicit chiral symmetry breaking LSB have been described in detail in references [38, 53]. To investigate the hadronic properties in the medium, we write the Lagrangian density within the chiral SU(3) model in the mean field approximation and determine the expectation values of the meson fields by minimizing the thermodynamical potential [54, 55].
III.
¯ MESONS IN THE MEDIUM D AND D
¯ We now examine the medium modifications for the D and D-meson masses in the asymmetric nuclear matter. The properties of nucleons and scalar mesons have been studied in the asymmetric hadronic matter within a chiral SU(3) model [39]. We assume that the additional effect of charmed particles in the medium leads to only marginal modifications [60] of these hadronic properties and do not need to be taken into account here. However, to investigate the medium modification of the D-meson mass, we need to know the interactions of the D-mesons with the light hadron sector. The light quark condensate has been shown to play an important role for the shift in the D-meson mass in the QCD sum rule calculations [15]. In the present chiral model, 5
the interactions to the scalar fields (nonstrange, σ and strange, ζ) as well as a vectorial Weinberg-Tomozawa interaction term modify the masses for D± mesons in the medium. These interactions were considered within the SU(3) chiral model to investigate the modifications of K-mesons in the dense (asymmetric) hadronic medium [38, 39, 40]. ¯ To consider the medium effects on the D and D-meson masses we generalize the chiral SU(3)-flavor model to include the charmed mesons. The scalar meson multiplet has now the expectation value √ 2 (σ + δ)/
hXi =
0
0
0
0 0 √ (σ − δ)/ 2 0 0
,
0
0
ζ 0 0 ζc
0
(2)
with ζc corresponding to the c¯c–condensate. The pseudoscalar meson field P can be written, including the charmed mesons, as
P =
where w =
√
2ζ/σ and wc =
π0 √ 2 −
π
2K − 1+w 2D 0 1+wc
π
+
2K + 1+w
¯0 2D 1+wc
π0
2K 0
2D −
− √2
1+w
1+wc
¯0 2K 1+w
0
0
2D + 1+wc
0
0
,
(3)
√
2ζc /σ. From PCAC, one gets the decay constants for the √ √ pseudoscalar mesons as fπ = −σ, fK = −(σ + 2ζ)/2 and fD = −(σ + 2ζc )/2. In the
present calculations, the value for the D-decay constant will be taken to be 135 MeV [19]. We note that for the decay constant of Ds+ , the Particle Data Group [61] quotes a value of fDs+ ≃ 200MeV . Taking a similar value also for fD would not affect our results qualitatively, however (see also [62]). The interaction Lagrangian modifying the D-meson mass can be written as [39] i h µ µ 0 ¯ 0 ) − (∂µ D 0 )D ¯ 0 + D + (∂µ D − ) − (∂µ D + )D − 3 p ¯ γ p + n ¯ γ n D (∂ D µ 8fD2 i ¯ 0 ) − (∂µ D 0 )D ¯ 0 − D + (∂µ D − ) − (∂µ D + )D − + p¯γ µ p − n ¯ γ µ n D 0 (∂µ D i √ m2D h ¯ 0 D 0 + (D − D + )) + δ(D ¯ 0 D 0 ) − (D − D + )) + (σ + 2ζc )(D 2fD √ 1 h ¯ 0 )(∂ µ D 0 ) + (∂µ D − )(∂ µ D + ) (σ + 2ζc ) (∂µ D − fD
LDN = −
6
¯ 0 )(∂ µ D 0 ) − (∂µ D − )(∂ µ D + ) + δ (∂µ D
i
d1 ¯ 0 )(∂ µ D 0 )) (¯ pp + n ¯ n)((∂µ D − )(∂ µ D + ) + (∂µ D 2 2fD d2 h ¯ 0 )(∂ µ D 0 ) + (∂µ D − )(∂ µ D + )) (¯ pp + n ¯ n))((∂µ D + 2 4fD i ¯ 0 )(∂ µ D 0 )) − (∂µ D − )(∂ µ D + )) + (¯ pp − n ¯ n)((∂µ D +
(4)
In (4) the first term is the vectorial interaction term obtained from the kinetic term in (1). The second term, which gives an attractive interaction for the D-mesons, is obtained from the explicit symmetry breaking term in (1). The third term arises from kinetic term of the pesudoscalar mesons [39, 40]. The fourth and fifth terms have been written down for the DN interactions, in analogy with the d1 and d2 terms in chiral SU(3) model [39, 40]. The last three terms in (4) represent the range term in the chiral model. It might be noted here that the interaction of the pseudoscalar mesons to the vector mesons, in addition to the pseudoscalar meson–nucleon vectorial interaction leads to a double counting in the linear realization of the chiral effective theory [63]. Within the nonlinear realization of the chiral effective theories, such an interaction does not arise in the leading or sub-leading order, but only as a higher order contribution [63]. Hence the vector meson-pesudoscalar interaction will not be considered within the present investigation. ¯ mesons yield the The Fourier transformations of the equations of motion for D and D dispersion relations, ω 2 + ~k 2 + m2D − Π(ω, |~k|) = 0,
(5)
¯ meson in the medium. where Π denotes the self energy of the D (D) Explicitly, the self energy Π(ω, |~k|) for the D meson doublet, (D 0 ,D + ) arising from the interaction (4) is given as i 1 h 3(ρ + ρ ) ± (ρ − ρ )) ω p n p n 4fD2 m2D ′ √ ′ + (σ + 2ζc ± δ ′ ) 2fD h d1 1 ′ √ ′ (σ + 2ζc ± δ ′ ) + 2 (ρps + ρns ) + − fD 2fD i d2 s s s s (ρ + ρ ) ± (ρ − ρ ) (ω 2 − ~k 2 ), + p n p n 4fD2
Π(ω, |~k|) =
7
(6)
where the ± signs refer to the D 0 and D + respectively. In the above, σ ′ (= σ − σ0 ), ζc ′ (= ζc − ζc 0 ) and δ ′ (= δ − δ0 ) are the fluctuations of the scalar-isoscalar fields σ and ζc , and
the third component of the scalar-isovector field, δ, from their vacuum expectation values. The vacuum expectation value of δ is zero (δ0 =0), since a nonzero value for it will break the isospin symmetry of the vacuum ( we neglect here the small isospin breaking effect arising from the mass and charge difference of the up and down quarks). In the above, ρp and ρn are the number densities of proton and neutron and ρs p and ρs n are their scalar densities. ¯ meson doublet, (D ¯ 0 ,D − ), the self-energy is calculated as Similarly, for the D i 1 h 3(ρp + ρn ) ± (ρp − ρn ) ω 2 4fD m2D ′ √ ′ + (σ + 2ζc ± δ ′ ) 2fD h d1 1 ′ √ ′ (σ + 2ζc ± δ ′ ) + 2 (ρps + ρns ) + − fD 2fD i d2 s s s s 2 ~2 + (ρ + ρ ) ± (ρ − ρ ) p n p n (ω − k ), 4fD2
Π(ω, |~k|) = −
(7)
where the ± signs refer to the D¯0 and D − respectively.
¯ mesons The optical potentials are calculated from the energies of the D and D U(ω, k) = ω(k) −
q
k 2 + m2D ,
(8)
¯ meson and ω(k) is the momentum dependent where mD is the vacuum mass for the D(D) ¯ meson. energy of the D(D) The parameters d1 and d2 are determined by a fit of the empirical values of the KN scattering lengths [64, 65, 66] for I=0 and I=1 channels [39, 40]. In the next section, we shall discuss the results for the D-meson mass modification obtained in the present effective chiral model as compared to the results in the literature, obtained from other approaches.
IV.
RESULTS AND DISCUSSIONS
¯ To study the D(D)-meson masses in asymmetric nuclear medium due to its interactions with the light hadrons, we have generalized the chiral SU(3) model used for the study 8
of the dense hadronic matter to SU(4) for the meson sector. The contributions from the various terms of the interaction Lagrangian (4) to the masses of the D ≡ (D 0 , D + ) and ¯ ≡ (D¯0 , D − ) are shown in figures 1 and 2, as functions of density. These are illustrated for D the isospin asymmetric case with the value of the asymmetry parameter, η = (ρn −ρp )/(2ρB ) as 0.5 and compared with the results obtained for the isospin symmetric case (η=0). The isospin symmetric part (∼ (ρn + ρp )) of the first term of (4), called the Weinberg-Tomozawa term, is attractive for D ≡ (D 0 , D + ) mesons and leads to a drop of the masses of the D +
¯ mesons in the nuclear medium, leads to and D 0 mesons, whereas it is repulsive for the D ¯ 0 and D − meson masses. In the isospin asymmetric nuclear medium, an increase of the D the Weinberg-Tomozawa term, leads to a mass splitting of the D 0 and D + mesons, giving a further drop in the mass of D + , whereas the asymmetry reduces the drop of the mass in D 0 . The D-meson self energy arising from the Weinberg-Tomozawa interaction, ΠW T (ω, |~k|) is given by the first term of equation (6), and at low densities, this turns out to be much smaller than (~k 2 + m2D ). One can then, as a first approximation replace ΠW T (ω, |~k|) by ΠW T (mD , |~k|) and solve for dispersion relation given by (5). Confining our attention to the Weinberg-Tomozawa interaction only, the energies of the D 0 and D + mesons, in the above approximation, are given by i 1 h ω(|~k|) ≃ (|~k|2 + m2D )1/2 − 2 3(ρp + ρn ) ∓ (ρp − ρn ) 8fD
(9)
One can note from the above equation (9) that at low densities, a given isospin asymmetry introduces equal increase (drop) for mass of D 0 (D + ) meson at low densities. However, at higher densities, there are deviations from the analytical expressions given by (9) as expected, though the qualitative feature of the D + (D 0 ) experiencing an attractive (repulsive) interaction from the vectorial Weinberg-Tomozawa term of (6) still remains the same, as can be seen from figure 1. One sees, from figure 2, that there is an increase in the masses of the D − and D¯0 and their mass shifts are equal for the case of isospin symmetric matter (η=0). However, in the presence of asymmetry, the D − mass is seen to have an increase whereas D¯0 mass drops in the asymmetric medium. These behaviours for D − and D¯0 can be understood by examining the asymmetric contributions of the Weinberg-Tomozawa term (the first term of (7)). The scalar meson exchange contribution to the self energy is given by the second term of 9
¯ mesons. Its interaction is equation (6) for D mesons and by the second term of (7) for D ¯ doublets for the isospin symmetric nuclear attractive and is identical for both the D and D matter (a negligible difference in the energies is due to the difference in their vacuum masses, mD+ = mD− =1869 MeV and mD0 = mD¯0 =1864.5 MeV). One might notice from the self energy terms due to the scalar meson exchange that a nonzero value of the scalar isovector δ-meson arising due to isospin asymmetry in nuclear matter, gives a drop in the masses for the D + and D − , whereas the δ contribution is repulsive for D 0 and D¯0 (σ ′ = σ − σ0 > 0 and δ ′ = δ < 0). This gives the D 0 mass to be higher than D + mass for asymmetric nuclear ¯ 0 ) to be higher than D − matter as seen in figure 1 and the mass of D¯0 mass (identical to D
mass (identical to D + mass) as plotted in figure 2. However, one might notice that the shifts in the D + (D − ) and D 0 (D¯0 ) masses about the isospin symmetric case (η=0) case are not equal and opposite. The reason for the seen asymmetry in the mass splittings is the following. For low baryon densities, one has σ ′ ∼ ρs ≃ ρB , and δ ′ ∼ (ρsp − ρsn ) ≃ (ρp − ρn ), so that one would expect the splittings of the masses of D + and D 0 to be symmetrical (about the
isospin symmetric matter) for a given baryon density ρB and isospin asymmetry parameter, η. However, these no longer hold good for higher densities and the mean field σ calculated in the isospin symmetric situation (η=0) turns out to be different for the asymmetric situation when the coupled equations of motions are solved for the scalar mean fields due to the presence of the scalar isovector δ field, as compared to when these equations are solved for symmetric nuclear matter (for η=0) in the absence of the δ meson. ¯ self energies due to range terms are given by the last The contributions to the D and D three terms of the right hand side of equations (6) and (7) respectively. The first term of the range terms given by the third term in (6) is repulsive whereas the second and third range terms have attractive contributions, when the isospin asymmetry is not taken into account. However, the isospin asymmetry, due to a nonzero value of the δ field, leads to an increase in the masses of the D + and D − mesons and a drop in the masses of D 0 and D¯0 from the isospin symmetric case. The second of the range terms (the d1 term) is attractive ¯0 and gives identical mass drops for D + and D 0 in the D doublet as well as for D − and D ¯ doublet. This term is proportional to (ρp + ρn ), which turns out to be different in the D s s for the isospin asymmetric case as compared to the isospin symmetric nuclear matter, due
10
to the presence of the δ meson. This is because the equations of motion for the scalar fields for the two situations (with/without δ mesons) give different values for the mean field, σ (∼ (ρs p + ρs n )). The last term of the range term (the d2 term) has a negative contribution for the energies of D + and D 0 mesons as well as for D − and D¯0 mesons for the isospin symmetric matter. The isospin asymmetric part arising from the ((ρs n − ρs p )) term of the
d2 term has a further drop in the masses for D ± mesons, whereas it increases the masses of the D 0 and D¯0 mesons from their isospin symmetric values. One sees from figures 1 and
2 that for both D + and D − mesons, the modification in the masses arising from the range terms, for the asymmetric matter (η=0.5), as compared to the isospin symmetric matter is negligible due to the increase from the first two range terms almost cancelling with the drop ¯ 0 mesons, the mass is seen to increase as compared to due to the d2 term. For the D 0 and D the isospin symmetric matter, due to the increase due to the second and third range terms dominating over the drop due to first range term. The values of the paramters d1 and d2 are fitted from the kaon-nucleon scattering lengths [38, 39] to be 2.56/mK and 0.73/mK respectively [40] and it is seen that the d2 term has a smaller contribution as compared to the d1 term. At high densities, these attractive d1 and d2 terms dominate over the first range ¯ mesons. There term (repulsive) and this leads to a decrease of the masses of the D and D is seen to be a substantial drop of D-meson masses at high densities due to the inclusion of this range term. The density dependence of the masses of the D mesons at specific values of the isospin asymmetric parameter, η, are shown in figure 3. The isospin asymmetry is seen to give a rise (drop) in the masses of the D 0 (D + ) as compared to their masses in the symmetric matter. For the isospin symmetric nuclear matter (η=0), the drop in the mass of the D + at ρB = ρ0 is about 81 MeV from its vacuum value of 1869 MeV and D − meson mass also has a drop of about 30 MeV, giving a mass splitting between the D + and D − mesons as about 51 MeV. A similar drop of the D-meson mass is also predicted by the QMC model [21] as well as ¯ mass in the QCD sum rule approach in [15]. In the recent for the isospin averaged D(D) QCD sum rule calculations [16], the mass drop for D + D − is about 50 MeV and the mass splitting is about 90 MeV at ρ = ρ0 , whereas for the isospin symmetric nuclear matter, we obtain these values as 110 MeV and 50 MeV respectively. The coupled channel calculations
11
[44] show a dominant modification of the D-meson width and only a small change in the D meson mass in the medium. When the DN interaction is taken to be a Tomozawa Weinberg interaction supplemented by a scalar-isoscalar interaction [48], the mass modification of the D mesons is seen to be around 10 MeV at ρ0 and about 50 MeV for a density of 2ρ0 . For the D − meson, there is an increase in the mass by about 10 MeV and 30 MeV at densities of ρ0 and 2ρ0 respectively. In the present investigation, the isospin dependence is seen to be quite prominent for D 0 as compared to D + . The isospin asymmetry in the medium is seen to give an increase in the D 0 mass, whereas it gives a drop for the D + mass as compared to the η=0 case. For the isospin asymmetry parameter, η=0.5, the D 0 mass is seen to rise by about 23 MeV and 105 MeV for ρB as ρ0 and 5ρ0 respectively from their values of 1783 MeV and 1441 MeV in isospin symmetric matter. For the same value of asymmetry parameter, η, the D + is seen to have a mass drop of about 18 MeV and 39 MeV for ρB = ρ0 and 5ρ0 respectively, from its η=0 values. This strong isospin dependence of the D mesons should show up in observables like their production as well as flow in asymmetric heavy ion collisions planned at the future facility at FAIR, GSI. Figure 4 shows the masses of the D − and D¯0 . It is seen that at high densities, both the D − and D¯0 are seen to have an increase in their masses in the asymmetric nuclear matter, as compared to the isospin symmetric case and this rise in the masses are seen to be similar for both D − and D¯0 . For example, at nuclear matter density, the masses of D − and D¯0 are 1838 MeV and 1840 MeV respectively for η=0.5, which are very similar to the values of 1839 MeV and 1834.5 MeV for the isospin symmetric matter. For ρB = 5ρ0 , the masses of D − and D¯0 are 1690 MeV and 1698 MeV respectively, which are higher by about 31 MeV and 43 MeV from the isospin symmetric case. One sees that the values of D − and D¯0 masses remain very similar at a given isospin asymmetry. But it is seen that the density effects on these masses are quite appreciable (a drop of about 30 MeV for nuclear matter density and of about 180 MeV for a density about five times nuclear matter density for η=0.5). ¯ doublets as functions of Figures 5 and 6 show the optical potentials for the D and D the momentum. These are shown for densities ρ0 and 5ρ0 . The isospin dependence of the optical potentials is seen to be quite significant for high densities for the doublet (D + , D 0),
12
as has also been seen in the case of their masses. But, we see the optical potentials for ¯ have very similar values for D − and D¯0 for a fixed value of the isospin asymmetric the D parameter. But, as already has been seen for the case of their masses, we see the optical potentials for both D − and D¯0 are quite different from the symmetric nuclear matter case at high densities. The decay widths of the charmonium states can be modified by the level crossings between ¯ creation due to the medium the excited states of J/Ψ (i.e., Ψ′ ,χc ) and the threshold for D D ¯ modifications of the D-meson masses [18]. In the vacuum, the resonances above the D D threshold, for example the Ψ′′ state, has a width of 25 MeV due to the strong open charm channel. On the other hand, the resonances below the threshold have a narrow width of ¯ a few hundreds of KeV, only. With the medium modification of the D(D)-meson masses, the channels for the excited states of J/Ψ, like χc , Ψ′ decaying to D + D − or D 0 D¯0 pairs can open up in the dense hadronic medium. This can increase the decay widths of Ψ′ and χc states at high densities. In figure 7, we show the density dependence of the mass of the D + D − as well as D 0 D¯0 pair, calculated in the present investigation. We see that the decay to D + D − channel is almost insensitive to isospin asymmetry. On the other hand, the decay channel to the D 0 D¯0 is seen to have a strong isospin asymmetry dependence, with the asymmetry shifting the onset of the decay to higher densities. The decay of charmonium ¯ has been studied in Ref. [18, 33]. It is seen to depend sensitively on the relative states to D D momentum in the final state. These excited states might become narrow [18] though the D-meson mass is decreased appreciably at high densities. It may even vanish at certain momenta corresponding to nodes in the wavefunction [18]. Though the decay widths for these excited states can be modified by their wave functions, the partial decay width of χc2 , due to absence of any nodes, can increase monotonically with the drop of the D + D − pair mass in the medium [18]. This can give rise to depletion in the J/Ψ yield in heavy ion ¯ pairs have also collisions. The dissociation of the quarkonium states (Ψ′ , χc , J/Ψ) into D D been studied [20, 67] by comparing their binding energies with lattice results on temperature dependence of heavy quark effective potential [37].
13
V.
SUMMARY
¯ To summarize we have investigated in a chiral model the in-medium masses of the D, Dmesons in asymmetric nuclear matter, arising due to their interactions with the nucleons and scalar mesons. The properties of the light hadrons – as studied in a SU(3) chiral model ¯ – modify the D(D)-meson properties in the dense hadronic medium. The SU(3) model with paramaters fixed from the properties of hadron masses in the vacuum, and low energy KN ¯ scattering data, is extended to SU(4) to derive the interactions of D(D)-mesons with the light hadron sector. The mass modifications for the D mesons are seen to be similar to earlier finite density calculations of QCD sum rules [16, 19] as well as to the quark meson coupling (QMC) model [21], in contrast to the small mass modifications in the coupled channel approach [44, 48]. In our calculations, the presence of the repulsive range term (the fourth term of (4) is compensated by the attractive d1 and d2 terms given by the last two terms in (4) and the latter (dominated by the d1 term) have an effect of reducing the masses ¯ mesons at high densities. of both D and D The medium modification of the D-masses can lead to a suppression in the J/Ψ- yield in heavy-ion collisions, since the excited states of J/Ψ and at a much higher density (≃ 5ρ0 ), ¯ pairs in the dense hadronic medium. The decay to D + D − pair seems J/Ψ can decay to D D to be insensitive to isospin dependence, whereas due to increase in the mass of D 0 D¯0 in the asymmetric medium, isospin asymmetry is seen to disfavour the decay of the charmonium states to D 0 D¯0 pair. The isospin dependence of D + and D 0 masses is seen to be dominant medium effect at high densities which might show in their production (D + /D0 ), whereas for the D − and D¯0 , one does see that even though these have a strong density dependence, their in-medium masses remain similar at a given value for the isospin asymmetry parameter η. ¯ meson optical The strong density dependence as well as the isospin dependence of the D(D) potentials in the asymmetric nuclear matter can be tested in the asymmetric heavy ion collision experiments at the future GSI facility [36].
14
Acknowledgments
We thank Arvind Kumar, Sambuddha Sanyal, Laura Tolos and J. Schaffner-Bielich for fruitful discussions. One of the authors (Amruta Mishra) is grateful to the Institut f¨ ur Theoretische Physik for warm hospitality and acknowledges financial support from Alexander von Humboldt Stiftung when this work was initiated. Financial support from Department of Science and Technology, Government of India (project no. SR/S2/HEP-21/2006) is also gratefully acknowledged.
[1] G. Agakichiev et al., CERES collaboration, Phys. Rev. Lett. 75, 1272 (1995); G. Agakichiev et al., CERES collaboration, Phys. Lett. B 422, 405 (1998); G. Agakichiev et al., CERES collaboration, Nucl. Phys. A661, 23c (1999). [2] N. Masera et al., HELIOS-3 collaboration, Nucl. Phys. A590, 93c (1995). [3] E. L. Bratkovskaya and W. Cassing, Nucl. Phys. A619, 413 (1997). [4] W. Cassing and E. L. Bratkovskaya, Phys. Rep. 308, 65 (1999). [5] A. Mishra, J. C. Parikh and W. Greiner, J. Phys. G 28, 151 (2002). [6] A. Mishra, J. Reinhardt, H. St¨ ocker and W. Greiner, Phys. Rev. C 66, 064902 (2002). [7] G. Q. Li, C. M. Ko and G. E. Brown, Nucl. Phys. A606, 568 (1996). [8] C. M. Ko, J. Phys. G 27, 327 (2001). [9] G. Q. Li, C.-H. Lee and G. E. Brown, Nucl. Phys. A625, 372 (1997). [10] S. Pal, C. M. Ko, and Z.-W. Lin, Phys. Rev. C 64, 042201 (2001). [11] F. Laue et al., Phys. Rev. Lett. 82, 1640 (1999). [12] C. Sturm et al., Phys. Rev. Lett. 86, 39 (2001). [13] A. F¨orster et al., KaoS Collaboration, J. Phys. G 28, 2011 (2002). [14] M. Menzel et al., Phys. Lett. B 495, 26 (2000). [15] A. Hayashigaki, Phys. Lett. B 487, 96 (2000). [16] T. Higler, R. Thomas, B. K¨ ampfer, nucl-th/0809.4996. [17] W. Liu, C. M. Ko and Z. W. Lin, Phys. Rev. C 65, 015203 (2001). [18] B. Friman, S. H. Lee and T. Song, Phys. Lett. B 548, 153 (2002).
15
Energy, (k=0) (MeV)
2000 D 1800 1600 total WT scalar range
1400 0
1
2
3
4
2000
Energy, (k=0) (MeV)
(a)
+
D
5
6
7
(b)
0
1800 1600 total WT scalar range
1400 0
1
2
3
4
5
6
7
B/ 0
FIG. 1: (Color online) The various contributions to D meson energies at zero momentum (for D+ in (a) and for D0 in (b)) in MeV plotted as functions of the baryon density in units of nuclear matter saturation density, ρB /ρ0 are shown for the isospin asymmetry parameter, η=0.5 and compared with the case of η=0 (dotted line).
16
Energy, (k=0) (MeV)
2000
(a)
1800
1600 total WT scalar range
1400 0
Energy, (k=0) (MeV)
2000
1
2
-
D
3
4
5
6
7
5
6
7
(b)
1800
1600 0
D
total WT scalar range
1400 0
1
2
3
4 B/ 0
¯ meson energies at zero momentum (for D − in FIG. 2: (Color online) The variuos contributions to D (a) and for D¯0 in (b)) in MeV plotted as functions of the baryon density in units of nuclear matter saturation density, ρB /ρ0 are shown for the isospin asymmetry parameter, η=0.5 and compared with the case of η=0 (dotted line).
17
Energy, (k=0) (MeV)
1900
=0 =0.1 =0.3 =0.5
1800 1700 +
D 1600 1500 1400
(a) 1300 0
1
2
3
4
5
6
7
Energy, (k=0) (MeV)
1900
=0 =0.1 =0.3 =0.5
1800 1700 0
D
1600 1500 1400
(b) 1300 0
1
2
3
4 B/
5
6
7
0
FIG. 3: (Color online) The D meson energies for zero momentum (k=0) (for D + in (a) and for D0 in (b)) in MeV plotted as functions of the baryon density in units of ρ0 , ρB /ρ0 for different values of the isospin asymmetry parameter, η. [19] P. Morath, W. Weise and S. H. Lee, 17th Autumn school: Lisbon 1999, QCD: Perturbative or Nonperturbative? Singapore: World Scientific, p. 425. [20] S. Digal, P. Petreczky and H. Satz, Phys. Lett. B 514, 57 (2001). [21] K. Tsushima, D. H. Lu, A. W. Thomas, K. Saito, and R. H. Landau, Phys. Rev. C 59, 2824
18
Energy, (k=0) (MeV)
1900 =0 =0.1 =0.3 =0.5
1800 1700 -
D 1600 (a) 1500
0
1
2
3
4
5
6
7
Energy, (k=0) (MeV)
1900 =0 =0.1 =0.3 =0.5
1800 1700
0
D
1600 (b) 1500
0
1
2
3
4
5
6
7
B/ 0
¯ mesons for zero momentum (for D − in (a) and for FIG. 4: (Color online) The energies of the D D¯0 in (b)) as functions of the baryon density in units of ρ0 (ρB /ρ0 ), are plotted for different values of the isospin asymmetry parameter, η. (1999); A. Sibirtsev, K. Tsushima, and A. W. Thomas, Eur. Phys. J. A 6, 351 (1999). [22] W. Cassing, E. L. Bratkovskaya, and A. Sibirtsev, Nucl. Phys. A691, 753 (2001). [23] Z. Lin and C. M. Ko, J. Phys. G 27, 617 (2001). [24] M. Gonin et al., NA50 Collaboration, Nucl. Phys. A610, 404c (1996).
19
0
-100
-100
U(k) (MeV)
U(k) (MeV)
0
+
-200
D
-300
B= 0
-400
0
250
500
750
-300
1000
0
250
500
750
1000
0
-100
+
D
-200 B=5 0
-300
=0 =0.3 =0.5
=0 =0.3 =0.5
(d) -100 0
U(k) (MeV)
(c)
U(k) (MeV)
=0 =0.3 =0.5
(b) -500
0
-400 -500
B= 0
-400
=0 =0.3 =0.5
(a) -500
0
D
-200
D -200
B=5 0
-300 -400
0
250 500 750 1000 Momentum, k (MeV)
-500
0
250 500 750 1000 Momentum, k (MeV)
FIG. 5: (Color online) The optical potentials (for D + and D 0 ) in MeV plotted as functions of the momentum, k at densities, ρ0 and 5ρ0 for different values of the isospin asymmetry parameter, η. [25] M. C. Abreu et al., NA50 Collaboration, Eur. Phys. J. C 14, 443 (200). [26] L. Rammello et al., NA50 Collaboration, Nucl. Phys. A638, 261c (1998); M. C. Abreu et al., Phys. Lett. B 450, 456 (1999); L. Ramello et al., Nucl. Phys. A715, 242 (2003). [27] T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986). [28] J. P. Blaizot and J. Y. Ollitrault, Phys. Rev. Lett. 77, 1703 (1996).
20
0
-50
-50
U(k) (MeV)
U(k) (MeV)
0
-
D
-100
B= 0
-150 -200
=0 =0.3 =0.5
(a) 0
250
500
750
0
D -100
-200
1000
=0 =0.3 =0.5
(b) 0
0
250
500
750
1000
0
-50
-
D
-100
B=5 0
-150
=0 =0.3 =0.5
=0 =0.3 =0.5
(d) -50
U(k) (MeV)
(c)
U(k) (MeV)
B= 0
-150
-200
0
D -100
B=5 0
-150 -200
0
250 500 750 1000 Momentum, k (MeV)
0
250 500 750 1000 Momentum, k (MeV)
FIG. 6: (Color online) The optical potentials (for D − and D¯0 ) in MeV plotted as functions of the momentum, k at densities, ρ0 and 5ρ0 for different values of the isospin asymmetry parameter, η. [29] B. Zhang, C. M. Ko, B. A. Li, Z. Lin, and B. H. Sa, Phys. Rev. C 62, 054905 (2000). [30] W. Cassing and E. L. Bratkovskaya, Nucl. Phys. A623, 570 (1997). [31] E. L. Bratkovskaya, W. Cassing and H. St¨ ocker, Phys. Rev. C 67, 054905 (2003). [32] L. Antoniazzi et al., E705 Collaboration, Phys. Rev. Lett. 70, 383 (1993); Y. Lemoigne et al., Phys. Lett. B 113, 509 (1982).
21
3800
=0, 0=0
=0 =0.5
’ 3600
c2 c1 c0
3400 +
-
D D
3200
J/ (a) 3000
0
1
2
3
4
5
3800
6
7
=0, 0=0
=0 =0.5
’ 3600
c2 c1
3400
c0 0
0
D D
3200
J/ (b) 3000
0
1
2
3
4
5
6
7
B/ 0
¯ pair (D + D − in (a) and D 0 D¯0 in (b)) in MeV plotted FIG. 7: (Color online) The mass of the DD as a function of ρB /ρ0 for values of the isospin asymmetry parameter, η as 0 amd 0.5. [33] Ye. S. Golubeva, E. L. Bratkovskaya, W. Cassing, and L. A. Kondratyuk, Eur. Phys. J. A 17, 275 (2003). [34] A. Sibirtsev, K. Tsushima, K. Saito and A. W. Thomas, Phys. Lett. B 484, 23 (2000). [35] S. H. Lee and C. M. Ko, Phys. Rev. C 67, 038202 (2003). [36] see e.g. http://www.gsi.de/GSI-future.
22
[37] F. Karsch, E. Laermann and A. Peikert, hep-lat/0012023; F. Karsch et al., Nucl. Phys. B 502, 321 (2001). [38] A. Mishra, E. L. Bratkovskaya, J. Schaffner-Bielich, S. Schramm and H. St¨ ocker, Phys. Rev. C 70, 044904 (2004). A. Mishra and S. Schramm, Phys. Rev. C 74, 064904 (2006). [39] A. Mishra, S. Schramm and W. Greiner, Phys. Rev. C 78, 024901 (2008). [40] Amruta Mishra, Arvind Kumar, Sambuddha Sanyal, S. Schramm, arXiv:0808.1937 (nucl-th). [41] A. Mishra, E. L. Bratkovskaya, J. Schaffner-Bielich, S.Schramm and H. St¨ ocker, Phys. Rev. C 69, 015202 (2004). [42] L.Tolos, J. Schaffner-Bielich and A. Mishra, Phys. Rev. C 70, 025203 (2004). [43] E.A. Veit, B. K. Jennings, A. W. Thomas, R. C. Berrett, Phys. Rev. D 31, 1033 (1985); L. Tolos, A. Ramos, A. Polls and T.T.S. Kuo, Nucl. Phys. A 690, 547 (2001); V. Koch, Phys. Lett. B 337, 7 (1994); P. B. Siegel and W. Weise, Phys. Rev. C 38, 2221 (1998). [44] L. Tolos, J. Schaffner-Bielich and H. St¨ ocker, Phys. Lett. B 635, 85 (2006). [45] J. Hofmann and M.F.M.Lutz, Nucl. Phys. A 763, 90 (2005). [46] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16, 255 (1966); Riazuddin and Fayyazuddin, Phys. Rev. 147, 1071 (1966). [47] T. Mizutani and A. Ramos, Phys. Rev. C 74, 065201 (2006). [48] L.Tolos, A. Ramos and T. Mizutani, Phys. Rev. C 77, 015207 (2008). [49] M.F.M. Lutz and C.L. Korpa, Phys. Lett. B 633, 43 (2006). [50] S.Weinberg, Phys. Rev. 166 1568 (1968). [51] S. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 2239 (1969); C.G. Callan, S. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 2247 (1969). [52] W. A. Bardeen and B. W. Lee, Phys. Rev. 177 2389 (1969). [53] P. Papazoglou, D. Zschiesche, S. Schramm, J. Schaffner-Bielich, H. St¨ ocker, and W. Greiner, Phys. Rev. C 59, 411 (1999). [54] D. Zschiesche, A. Mishra, S. Schramm, H. St¨ ocker and W. Greiner, Phys. Rev. C 70, 045202 (2004). [55] A. Mishra, K. Balazs, D. Zschiesche, S. Schramm, H. St¨ ocker, and W. Greiner, Phys. Rev. C 69, 024903 (2004).
23
[56] J. J. Sakurai, Currents and Mesons, University of Chicago Press, Chicago, 1969. [57] O. Dumbrajs, R. Koch, L. Pilkuhn, G. C. Oades, H. Behrens, J. J. De Swart and P. Kroll, Nucl. Phys. B 216, 277 (1983). [58] Th. A. Rijken, V. G. J. Stoks, Y. Yamamoto, Phys. Rev. C 59, 21 (1999); V. G. J. Stoks and Th. A. Rijken, Phys. Rev. C 59, 3009 (1999). [59] G. H¨ohler, E. Pietarinen, I. Sabba-Stefanescu, F. Borkowski, G. G. Simon, V. H. Walther, R. D. Wendling, Nucl. Phys. B 114, 505 (1976). [60] D. R¨ oder, J. Ruppert, and D.H. Rischke, Phys. Rev. D 68, 016003 (2003). [61] K. Hagiwara et al, Phys. Rev. D 66, 010001 (2002). [62] T-W Chiu and T-H Hsieh, hep-lat/0305016). [63] B.Borasoy and U-G. Meissner, Int. J. Mod. Phys. A 11 5183 (1996). [64] G. E. Brown, C-H Lee, M. Rho, and V. Thorsson, Nucl. Phys. A567, 937 (1994). [65] J. Schaffner-Bielich, I. N. Mishustin, J. Bondorf, Nucl. Phys. A 625, 325 (1997). [66] T. Barnes and E. S. Swanson, Phys. Rev. C 49, 1166 (1994). [67] C. Y. Wong T. Barnes, E. S. Swanson and H. W. Crater, nucl-th/0112023.
24
